E0022 theorem notation E0020 → E0001

Fréchet iterated Faà di Bruno

The discrete iterated Faà di Bruno E0011 passes to Fréchet derivatives via Taylor composition E0025: the covering sums collapse to partition sums, and the iterated increments \(\Delta^\kappa\) become iterated differentials \(D^\kappa\) E0021.

(1) Definition (Partition grouping coefficient). For \(\gamma \in \IN_0^S\) and \(\kappa \in \KM_+^m(S)\), the partition grouping coefficient is

\[ \Part_m(\gamma, \kappa) := \#\set{H \in \Part_m(S(\gamma)) : \nu(H) = \kappa}, \]

the number of \(m\)-fold partitions of the Boolean realization \(S(\gamma)\) whose higher profile map \(\nu\) E0007 equals \(\kappa\). The coefficient is zero unless \(\kappa \vdash \gamma\) (i.e. \(\lf(\kappa) = \gamma\)). For \(m = 2\) and \(\kappa \in \KM_+(S)\), this reduces to the multinomial-type coefficient \(\Part_2(\gamma, \kappa) = \gamma! / (\kappa! \prod_\alpha (\alpha!)^{\kappa(\alpha)})\).

(2) Theorem. Let \(X_0, \dots, X_m\) be Banach spaces, let \(f_r: X_{r-1} \to X_r\) be \(C^n\) near \(x_{r-1}\), and put \(x_0 = x\), \(x_r = f_r(x_{r-1})\), \(z = x_m\). Fix directions \(v_1, \dots, v_k \in X_0\) and let \(\gamma \in \IN_0^k\) with \(1 \leq |\gamma| \leq n\).

  • Faà di Bruno:

    \[ D^\gamma(f_m \circ \cdots \circ f_1;\, x;\, v_\bullet) = \sum_{\kappa \in \KM_+^m(k)} \Part_m(\gamma, \kappa)\, D^\kappa(f_1, \dots, f_m;\, x;\, v_\bullet). \]

  • Taylor composition:

    \[ (f_m \circ \cdots \circ f_1) (x + {\textstyle\sum_i} t_i v_i) = z + \sum_{\substack{0 < |\gamma| \leq n \\ \kappa \in \KM_+^m(k)}} \frac{t^\gamma}{\gamma!}\, \Part_m(\gamma, \kappa)\, D^\kappa(f_1, \dots, f_m;\, x;\, v_\bullet) + o(|t|^n). \]

    The sum is finite: \(\Part_m(\gamma, \kappa) = 0\) unless \(\kappa \vdash \gamma\), and only derivatives of order \(\leq |\gamma| \leq n\) appear.

Proof. Write \(P_r = T^n(f_r; x_{r-1})\) for the order-\(n\) Taylor polynomials and \(F = f_m \circ \cdots \circ f_1\). By Taylor composition E0025 applied \(m - 1\) times, \(T_*^n(F; x) = \pi_{\leq n}(P_m \circ \cdots \circ P_1 - z)\), so \(F(x + h) = z + \pi_{\leq n}(P_m \circ \cdots \circ P_1 - z)(h) + o(\|h\|^n)\). The truncated composite \(Q := \pi_{\leq n}(P_m \circ \cdots \circ P_1 - z)\) is a polynomial map between Banach spaces — in particular a map between abelian groups — so the discrete iterated Faà di Bruno E0011 applies to the chain \(P_1, \dots, P_m\).

The discrete formula sums over \(m\)-fold coverings \(K\) of the slot set, but only partitions survive the passage to derivatives. Grade every term by its total degree in the slot directions. Each term of a difference \(\Delta(P; x'; w_1, \dots, w_p)\) of a polynomial map contains every direction \(w_i\) at least once, since the alternating sum kills all monomials missing some \(w_i\); inductively, every term of \(\Delta^K\) contains each slot direction at least as often as the slot occurs among the leaves of \(K\), so all terms have slot degree \(\geq \mathrm{wt}(K)\). On a slot set of size \(N\) the weight bound E0008 gives \(\mathrm{wt}(K) \geq N\), with equality exactly for \(K \in \Part_m\). Hence only partitions contribute terms that are multilinear in the \(N\) slot directions, and for \(H \in \Part_m\) the multilinear part of \(\Delta^H\) is the levelwise polarization \(D^H(f_1, \dots, f_m; x; \cdot)\) E0021: under \(p\) differences a homogeneous part of degree \(j\) vanishes for \(p > j\), polarizes exactly to \(D^j\) for \(p = j\), and for \(p < j\) leaves only terms of slot degree \(> p\), which are not multilinear.

Ad Faà di Bruno) By E0020 and Taylor coefficient identification E0024, \(D^\gamma(F; x; v_\bullet) = D^{|\gamma|}(Q; 0; u)\) with slot directions \(u = v \circ \pi\) on \(S(\gamma)\), and \(D^{|\gamma|}(Q; 0; u)\) is the multilinear part of \(\Delta(Q; 0; u)\). Expanding \(\Delta(Q; 0; u)\) by the discrete formula on \(S(\gamma)\) and extracting multilinear parts leaves \(\sum_{H \in \Part_m(S(\gamma))} D^H\) by the collapse above. \(D^H\) depends only on the profile \(\kappa = \nu(H)\) by symmetry of the differentials (as in profile invariance E0010). Grouping the \(\#\set{H \in \Part_m(S(\gamma)) : \nu(H) = \kappa} = \Part_m(\gamma, \kappa)\) partitions with the same profile gives the stated sum.

Ad Taylor composition) Restrict to \(h = \sum_i t_i v_i\). Since \(\|h\| \leq C|t|\), the Peano remainder is \(o(|t|^n)\). The polynomial part \(\pi_{\leq n}(P_m \circ \cdots \circ P_1 - z)(\sum t_i v_i)\) is a polynomial of degree \(\leq n\) in the \(t_i\). By the multivariate Taylor formula E0024, its \(t^\gamma\)-coefficient is \(D^\gamma(F; x; v_\bullet)/\gamma!\); substituting the Faà di Bruno formula gives the stated expansion.

(3) Validation (AI review, 2026-07-19, claude-fable-5, pass). Statement verified (classical cases \(m = 1\); \(m = 2\) chain rule; \(\Part_2\) closed form; numeric check \(n = 2\), \(m = 2\) against the truncated composite). The covering→partition collapse argument as previously written was invalid: increments of non-partition coverings do not vanish under degree-\(\leq n\) truncation (counterexample \(n = 4\), \(H = \set{\set{1}, \set{1,2}}\)). Replaced with the slot-degree grading argument via the weight bound E0008 and multilinear extraction; added E0008, E0024 to dependencies. Statement unchanged.

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